Results 1 to 10 of about 28,795 (128)
Uniform convergence guarantees for the deep Ritz method for nonlinear problems. [PDF]
Adv Contin Discret Model, 2022 AbstractWe provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover nonlinear variational problems such as the p-Laplace equation or the Modica–Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across bounded families of
Dondl P, Müller J, Zeinhofer M.
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Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions [PDF]
Communications in Computational Physics, 2021 We propose a new method to deal with the essential boundary conditions encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are non-interpolatory, which makes the enforcement of the essential boundary conditions a nontrivial matter.
Liao, Yulei, Ming, Pingbing
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Solving a Class of High-Order Elliptic PDEs Using Deep Neural Networks Based on Its Coupled Scheme
Mathematics, 2022 Deep learning—in particular, deep neural networks (DNNs)—as a mesh-free and self-adapting method has demonstrated its great potential in the field of scientific computation. In this work, inspired by the Deep Ritz method proposed by Weinan E et al.
Xi’an Li +3 more
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Convergence Rate Analysis for Deep Ritz Method
Communications in Computational Physics, 2022 Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) \cite{wan11} for second order elliptic equations with Neumann boundary conditions.
Duan, Chenguang +5 more
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Variational methods and deep Ritz method for active elastic solids
Soft Matter, 2022 Variational and deep-Ritz (DR) methods for active elastic solids with applications in the morphogenesis of cell monolayer: (A) Spontaneous bending predicted using DR learning method, (B) Gravitaxis: spontaneous bending with and without gravity.
Haiqin Wang +4 more
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Error analysis of deep Ritz methods for elliptic equations
Analysis and Applications, 2023 Using deep neural networks to solve partial differential equations (PDEs) has attracted a lot of attention recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on the deep Ritz method (DRM) for second-order elliptic equations with Dirichlet, Neumann ...
Yuling Jiao +4 more
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Deep Ritz method with adaptive quadrature for linear elasticity
Computer Methods in Applied Mechanics and Engineering, 2023 In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the $H^{1/2}(Γ_D)$ norm is introduced and analyzed for linear elasticity equation in order to deal with the (essential) Dirichlet boundary condition. We show that the resulting deep Ritz method
Min Liu, Zhiqiang Cai, Karthik Ramani
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Robin Pre-Training for the Deep Ritz Method
Proceedings of the Northern Lights Deep Learning Workshop, 2023 We analyze the training process of the Deep Ritz Method for elliptic equations with Dirichlet boundary conditions and highlight problems arising from essential boundary values. Typically, one employs a penalty approach to enforce essential boundary conditions, however, the naive approach to this problem becomes unstable for large penalizations. A novel
Courte, Luca, Zeinhofer, Marius
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Deep Ritz Methods for Laplace Equations with Dirichlet Boundary Condition
CSIAM Transactions on Applied Mathematics, 2022 Summary: Deep Ritz methods (DRM) have been proven numerically to be efficient in solving partial differential equations. In this paper, we present a convergence rate in \(H^1\) norm for deep Ritz methods for Laplace equations with Dirichlet boundary condition, where the error depends on the depth and width in the deep neural networks and the number of ...
Duan, Chenguang +5 more
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Deep convolutional Ritz method: parametric PDE surrogates without labeled data
Applied Mathematics and Mechanics, 2023 AbstractThe parametric surrogate models for partial differential equations (PDEs) are a necessary component for many applications in computational sciences, and the convolutional neural networks (CNNs) have proven to be an excellent tool to generate these surrogates when parametric fields are present.
Fuhg, J. N. +4 more
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